Ask Dr. Math

Middle School Archive

Browse Middle School Puzzles
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to frequently posed puzzles:
    Letter+number puzzles.
    Number sentences.
    Remainder/divisibility puzzles.

    1000 lockers.
    Getting across the river.
    Heads, legs: how many animals?
    A hen and a half...
    How many handshakes?
    Last one at the table.
    Measuring with two containers.
    Monkeys dividing coconuts.
    Squares in a checkerboard.
    Weighing a counterfeit coin.
    What color is my hat?

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Impossible Word [04/14/2001]

What word comes out to 575,151 when you multiply together the values of its letters?

Indefinite Series, Perfect Squares [01/01/1998]

Across the first row of an 11-column table are the numbers 1991, 1992, 1993, .... 2000, 2001.

Jack is Older than Jill [11/16/2001]

If you reverse the digits in Jack's age, you get Jill's age. The sum of their ages equals 11 times the difference between them. Jack is older than Jill. What are their ages?

Join the Dots [09/02/2001]

Given 9 dots in a square, how can you connect them with only 4 lines without picking up your pencil or going through a dot more than once?

Knights of the Round Table [07/01/1998]

If x knights are sitting at a round table, and every other one is removed, who is the last one left sitting at the table?

Largest Magic Square Ever Known [09/18/2001]

What is the largest magic square ever constructed?

Laying Eggs Better by Half [12/05/2001]

If a hen and a half lays an egg and a half in a day and a half, how many and a half that lay better by half will lay half a score and a half in a week and a half?

Least Perimeter [10/13/2002]

Finding a formula for least perimeter of a square or rectangle.

Left-Sided Rhombuses in a Larger Rhombus [05/22/2000]

How many left-sided, right-sided, and vertical rhombuses can be found in a larger NxN rhombus?

Links from 1 to 63 [02/19/2002]

You are given 1 chain of 63 links to use instead of money. Your task is to cut the chain in 3 places so that you'll be able to hand a person any number of links he asks for, from 1-63.

Longest Roman Numeral [05/03/2007]

Which year in our recorded history has had the greatest number of Roman numerals in it?

Magic Square [5/16/1996]

How can I make a magic square?

Magic Square Variations [08/29/2001]

How many ways are there to arrange the first nine counting numbers?

Magic Star Puzzle [9/2/1996]

I have a star puzzle shaped like 2 triangles with 4 circles in each row. We have to use the integers 1 to 12 and the sum of each row must be the same...

Magic Triangle, 2 Numbers/Side [04/25/2003]

Natural numbers can be placed in an arrangement so that the sum of 3 numbers on each side of a triangle is always the same. Is it possible to find a magic triangle with 2 numbers?

Magic Triangle Puzzle [09/24/2001]

The numbers in the squares are the sums of the numbers in the circles to which they are connected. Discover a method to help you find the numbers to put in the circles.

Magic Triangle Sums [02/04/2003]

Find 6 consecutive odd numbers such that their magic triangle sum is 25. Find 6 consecutive numbers such their magic triangle sum is 4.

Magic Wheel [03/20/2002]

Given a wheel with 16 spokes and a hub, enter the whole numbers 1-17 in the hub and at the end of spokes. The sums of the 3 numbers along the spokes must be equal.

Make 24 [01/03/2003]

We have to make 24 by using 1, 3, 4, and 6.

Make $5 Using One of Each Coin [07/07/2001]

You have 100 coins: pennies, nickels, dimes, quarters, and half dollars. Use at least one of each to add up to $5.00.

Make 7,7,3,3 into 24 Using + - * and / [02/12/2002]

Make the numbers 7,7,3,3 equal 24 using the operations addition, subtraction, multiplication, and division.

Making 5+5+5=550 a True Statement [9/30/1995]

5+5+5=550. Add one line to this to make this statement true. It must be a horizontal, vertical or diagonal line and it cannot go through the = or the +'s.

Making $5 Using 50 Coins [12/02/2005]

How many ways can you make $5 with 50 coins and without using dimes?

Making a Hard Square Sum Problem Easier [01/30/2004]

Given these eight numbers: 1/4, 1/2, 3/4, 1, 1 1/4, 1 1/2, 1 3/4, and 2, place three numbers along each side of a square so that the sum of the three numbers on each side of the square is equal to 3. Is there a trick to doing this kind of problem?

Making a Magic Square [7/4/1996]

How do you make a magic square?

Making a Parallelogram [06/19/2001]

Make five shapes - a parallelogram, a square, and three triangles - into one parallelogram.

Making Change [04/15/1999]

What is the greatest amount of money John can have in his pocket?

Making Change for a Dollar [05/29/2001]

What is the smallest number of coins you CANNOT make change with?

Matching Pairs of Socks [11/1/1995]

In your sock drawer, you have 10 blue socks, 12 white socks, and 6 red socks that are all mixed together. It is dark and you can't see the colors of your socks. What is the least number of socks that you have to pull out to make sure that you will have a matching pair?

Math Puzzle: Day of the Week [02/04/1998]

Why does the following math problem work, irrespective of the number you use 0-7 for the day of the week?

Math Tricks [1/14/1996]

A student asks how math tricks involving multiplying and dividing big numbers in your head are done.

McNuggets [03/10/2002]

At one McDonald's location, McNuggets come in boxes of 6, 9, and 20. What would be the largest number of McNuggets that you could not buy?

Milk Bottle Puzzle [09/24/2001]

Given 18 milk bottles and a milk crate 6 holes wide by 4 holes deep, put an even number of bottles into every row and column.

Million Point Word Puzzle [05/18/2005]

If a = 1, b = 2, ... , z = 26, is there a word that when the values of the letters are multiplied will make a product of one million?

The Missing Dollar [03/29/1997]

Three men went to a restaurant for dinner and spent $30, but the cashier gave the waiter back $5. The waiter kept $2...

Monkeys Dividing the Coconut Pile [07/28/1998]

A monkey divides a pile of coconuts into three, tosses the extra one, and takes his share...

More Monkeys and Nuts [06/07/1999]

How can I solve this variation on the monkeys and the nuts puzzle?

Multiples of Four [10/22/2001]

How many multiples of four from 1 through 1000 do not contain any of the digits 6, 7, 8, 9, or 0?

Multiplying/Adding Fractions Gives Same Answer [03/01/2002]

Find two fractions which, when multiplied and added, give the same answer.

Multiplying a Number by 4 [01/11/1999]

Can you help me find a few numbers that when you multiply them by 4, you will get the original number backwards?

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